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@ -2,7 +2,7 @@
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#+begin_src jupyter-python
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%display latex
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var("G, phi, gamma, delta, t, a, Omega", domain=RR)
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var("G, phi, gamma, delta, t, a, b, c, d, Omega", domain=RR)
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var("z", domain=CC)
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#+end_src
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@ -52,3 +52,24 @@ t\right)} \sin\left(-\delta t - \phi\right)\]
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\Omega a + z^{2}} \\ -\frac{\Omega + a}{\Omega^{2} + \Omega a + z^{2}} &
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\frac{z}{\Omega^{2} + \Omega a + z^{2}} \end{array}\right)\]
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:END:
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#+begin_src jupyter-python
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matrix([[0, 0], [1, 0]]) * matrix([[a, b], [c, d]])
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#+end_src
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#+RESULTS:
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:RESULTS:
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\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} 0 & 0 \\ a &
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b \end{array}\right)\]
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:END:
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#+begin_src jupyter-python
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matrix([[0, 1], [-1, 0]]) * matrix([[a, b], [c, d]])
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#+end_src
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#+RESULTS:
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:RESULTS:
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\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} c & d \\ -a
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& -b \end{array}\right)\]
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:END:
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@ -15,7 +15,7 @@
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A = np.array([[0, Ω], [-Ω, 0]])
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η = 2
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ω_c = 1
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t_max = 50
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t_max = 16
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def α(t):
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@ -491,7 +491,7 @@ There remains the zero point energy.
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** Exponential Fit
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First we need an exponential fit for our BCF.
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#+begin_src jupyter-python
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W, G_raw = utilities.fit_α(α, 5, 80, 10_000)
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W, G_raw = utilities.fit_α(α, 3, 80, 10_000)
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τ = np.linspace(0, t_max, 1000)
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#+end_src
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@ -508,7 +508,7 @@ Looks quite good.
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#+RESULTS:
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:RESULTS:
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| hline | <AxesSubplot:> |
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[[file:./.ob-jupyter/e0b65e6c0c295f2ced7f7e42578fe0416afc6475.svg]]
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[[file:./.ob-jupyter/be47577bc9cef984dc52c230eb81af91e1b76fa6.svg]]
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:END:
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** Calculate the Magic Numbers
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@ -519,7 +519,7 @@ We begin with the $\varphi_n$ and $G_n$ from the original ~G~.
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#+end_src
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#+RESULTS:
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: array([ 2.1834114 , 1.06741778, 2.81544667, -0.46637419, -2.28665296])
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: array([-1.18710245, 0.64368323, 2.11154332])
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#+begin_src jupyter-python
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G = np.abs(G_raw)
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@ -527,7 +527,7 @@ We begin with the $\varphi_n$ and $G_n$ from the original ~G~.
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#+end_src
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#+RESULTS:
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: array([0.09682231, 0.46421051, 0.00678969, 0.69850585, 0.22961865])
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: array([0.51238635, 0.62180167, 0.10107935])
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Now we calculate the real and imag parts of the ~W~ parameters and
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call them $\gamma_n$ and $\delta_n$.
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@ -555,7 +555,7 @@ this is not the case.
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#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
: array([-0.28690071, -1.23300496, -0.0505763 , 1.06808563, -6.88715892])
|
||||
: array([-1.19725058, -2.0384181 , -0.23027627])
|
||||
|
||||
Now the \(z_k\) the roots of \(\delta_k^2 + (\gamma_k + z)^2\). *We
|
||||
don't include the conjugates.*
|
||||
|
|
@ -565,9 +565,8 @@ don't include the conjugates.*
|
|||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
: array([-0.34633391-0.08456407j, -0.98090077-0.4577907j ,
|
||||
: -0.0725635 -0.00743661j, -2.01573795-1.55243654j,
|
||||
: -3.36022865-4.05504392j])
|
||||
: array([-2.29230292-2.71252483j, -1.07996581-0.719112j ,
|
||||
: -0.28445344-0.09022829j])
|
||||
|
||||
** Construct the Polynomials
|
||||
#+begin_src jupyter-python
|
||||
|
|
@ -629,12 +628,10 @@ And find its roots.
|
|||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
: array([-3.36385705-4.0499244j , -3.36385705+4.0499244j ,
|
||||
: -1.98189805-1.48637768j, -1.98189805+1.48637768j,
|
||||
: -0.86007885-0.39886909j, -0.86007885+0.39886909j,
|
||||
: -0.30506379-0.10051864j, -0.30506379+0.10051864j,
|
||||
: -0.19364497-1.48889586j, -0.19364497+1.48889586j,
|
||||
: -0.07122206-0.01054089j, -0.07122206+0.01054089j])
|
||||
: array([-2.28139877-2.68284887j, -2.28139877+2.68284887j,
|
||||
: -0.93979297-0.62025112j, -0.93979297+0.62025112j,
|
||||
: -0.24204514-0.10390896j, -0.24204514+0.10390896j,
|
||||
: -0.19348529-1.49063362j, -0.19348529+1.49063362j])
|
||||
|
||||
Let's see if they're all unique. This should make things easier.
|
||||
#+begin_src jupyter-python
|
||||
|
|
@ -647,73 +644,117 @@ Let's see if they're all unique. This should make things easier.
|
|||
Very nice!
|
||||
|
||||
** Calculate the Residuals
|
||||
These are the prefactors for the diagonal.
|
||||
#+begin_src jupyter-python
|
||||
R_l = f_0(master_roots) / p.deriv()(master_roots)
|
||||
R_l
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
: array([ 0.00019854+1.13582372e-04j, 0.00019854-1.13582372e-04j,
|
||||
: 0.01068261+2.14137130e-03j, 0.01068261-2.14137130e-03j,
|
||||
: 0.04932196+2.54768525e-02j, 0.04932196-2.54768525e-02j,
|
||||
: 0.02800713-3.28511946e-03j, 0.02800713+3.28511946e-03j,
|
||||
: -0.08930968+3.64119054e-01j, -0.08930968-3.64119054e-01j,
|
||||
: 0.00109945-2.04813251e-03j, 0.00109945+2.04813251e-03j])
|
||||
: array([ 0.00251589-0.00080785j, 0.00251589+0.00080785j,
|
||||
: 0.06323421+0.02800137j, 0.06323421-0.02800137j,
|
||||
: 0.02732669-0.00211665j, 0.02732669+0.00211665j,
|
||||
: -0.09307679+0.36145133j, -0.09307679-0.36145133j])
|
||||
|
||||
And these are for the most compliciated element.
|
||||
#+begin_src jupyter-python
|
||||
R_l_21 = (Ω + α_tilde(master_roots))* f_0(master_roots) / p.deriv()(master_roots)
|
||||
R_l_21
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
: array([-0.00325014-0.02160514j, -0.00325014+0.02160514j,
|
||||
: 0.00074814-0.05845205j, 0.00074814+0.05845205j,
|
||||
: -0.00094159-0.00084894j, -0.00094159+0.00084894j,
|
||||
: 0.00344359+0.56219924j, 0.00344359-0.56219924j])
|
||||
|
||||
|
||||
Now we can calculate \(G\).
|
||||
#+begin_src jupyter-python
|
||||
def G_12_ex(t):
|
||||
t = np.asarray(t)
|
||||
return Ω * (R_l[None, :] * np.exp(t[:, None] * master_roots[None, :])).real.sum(axis=1)
|
||||
return Ω * (R_l[None, :] * np.exp(t[:, None] * master_roots[None, :])).real.sum(
|
||||
axis=1
|
||||
)
|
||||
|
||||
|
||||
def G_11_ex(t):
|
||||
t = np.asarray(t)
|
||||
return (R_l[None, :] * master_roots[None, :] * np.exp(t[:, None] * master_roots[None, :])).real.sum(axis=1)
|
||||
return (
|
||||
R_l[None, :]
|
||||
,* master_roots[None, :]
|
||||
,* np.exp(t[:, None] * master_roots[None, :])
|
||||
).real.sum(axis=1)
|
||||
|
||||
def G_12_ex_alt(t):
|
||||
|
||||
def G_21_ex(t):
|
||||
t = np.asarray(t)
|
||||
return (R_l[None, :] * master_roots[None, :]**2 * np.exp(t[:, None] * master_roots[None, :])).real.sum(axis=1) / Ω
|
||||
return -(R_l_21[None, :] * np.exp(t[:, None] * master_roots[None, :])).real.sum(
|
||||
axis=1
|
||||
)
|
||||
|
||||
|
||||
def G_21_ex_alt(t):
|
||||
t = np.asarray(t)
|
||||
return (
|
||||
R_l[None, :]
|
||||
,* master_roots[None, :] ** 2
|
||||
,* np.exp(t[:, None] * master_roots[None, :])
|
||||
).real.sum(axis=1) / Ω
|
||||
|
||||
|
||||
def G_ex(t):
|
||||
t = np.asarray(t)
|
||||
if t.size == 1:
|
||||
t = np.array([t])
|
||||
diag = G_11_ex(t)
|
||||
return (
|
||||
np.array([[diag, G_12_ex(t)], [G_21_ex(t), diag]]).swapaxes(0, 2).swapaxes(1, 2)
|
||||
)
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
|
||||
#+begin_src jupyter-python
|
||||
plt.plot(τ, G_11_ex(τ))
|
||||
plt.plot(τ, G_12_ex_alt(τ))
|
||||
plt.plot(ts, proper @ np.array([1, 0]))
|
||||
plt.plot(τ, G_ex(τ).reshape(len(τ), 4))
|
||||
plt.plot(ts, proper.reshape(len(ts), 4), linestyle="--")
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
:RESULTS:
|
||||
| <matplotlib.lines.Line2D | at | 0x7f338516e280> | <matplotlib.lines.Line2D | at | 0x7f338516e2b0> |
|
||||
[[file:./.ob-jupyter/dd87bd9601ab3679ef0a16610edaae5a0d8bba93.svg]]
|
||||
| <matplotlib.lines.Line2D | at | 0x7f33843671f0> | <matplotlib.lines.Line2D | at | 0x7f33843675b0> | <matplotlib.lines.Line2D | at | 0x7f3384367a00> | <matplotlib.lines.Line2D | at | 0x7f3384370ee0> |
|
||||
[[file:./.ob-jupyter/748498e7edf248e20e45b9b326b840d531aee592.svg]]
|
||||
:END:
|
||||
|
||||
G12 is still a bit broken... Funnily i can get it to work by "knowing"
|
||||
how it should look like. With this, it looks like the numerical
|
||||
solution :).
|
||||
|
||||
#+begin_src jupyter-python
|
||||
def α_tilde(z):
|
||||
return (-G * ((z + γ) * s + δ * c)/ (δ**2 + (γ + z)**2)).sum()
|
||||
return (
|
||||
-G[None, :]
|
||||
,* ((z[:, None] + γ[None, :]) * s[None, :] + δ[None, :] * c[None, :])
|
||||
/ (δ[None, :] ** 2 + (γ[None, :] + z[:, None]) ** 2)
|
||||
).sum(axis=1)
|
||||
|
||||
|
||||
def f_test(z):
|
||||
return Ω**2 + (Ω * (-G * ((z + γ) * s + δ * c)/ (δ**2 + (γ + z)**2))).sum() + z**2
|
||||
mpmath.findroot(f_test, Ω, solver='muller')
|
||||
return (
|
||||
Ω ** 2
|
||||
+ (Ω * (-G * ((z + γ) * s + δ * c) / (δ ** 2 + (γ + z) ** 2))).sum()
|
||||
+ z ** 2
|
||||
)
|
||||
|
||||
|
||||
mpmath.findroot(f_test, Ω, solver="muller")
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
: mpc(real='-0.19364496699331084', imag='1.4888958605756884')
|
||||
: mpc(real='-0.19348529127068425', imag='1.4906336228104743')
|
||||
|
||||
#+begin_src jupyter-python
|
||||
plt.plot(τ, G_12_ex_alt(τ))
|
||||
plt.plot(τ, -G_12_ex(τ))
|
||||
plt.plot(τ, G_21_ex_alt(τ) - G_21_ex(τ))
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
:RESULTS:
|
||||
| <matplotlib.lines.Line2D | at | 0x7f338505c880> |
|
||||
[[file:./.ob-jupyter/069324aef03627d38595b0ca71126582c8766460.svg]]
|
||||
| <matplotlib.lines.Line2D | at | 0x7f337f9052b0> |
|
||||
[[file:./.ob-jupyter/e70ece655bc2da46d606acf9b0cd1ca6accfb5a9.svg]]
|
||||
:END:
|
||||
|
||||
WEIRD!
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue